私立中原大學八十八學年度研究所招生考試命題紙

所組別 :資訊工程學系 科目 :計算機數學 考試時間 : 05月 01日 第 2 節


1.
(a).[6%] Construct an LU decomposition for the matrix
.
(b).[6%] Find the determinant of A.
(c).[10%] Find .

2. An matrix P  is called an orthogonal matrix if P  is invertible and.
Let .

(a).[6%]Show that the columns of P  are orthonormal.
(b).[8%]Find the eigenvalues of A and corresponding eigenvectors.
(c).[6%]Find an orthogonal matrix P  such that is diagonal.

3.
(a).[6%] Find the ordinary generating function of the sequence

(b).[6%] Evaluate the sum .

4.
(a).[6%] Evaluate  .
(b).[6%] Let  . Derive a recurrence relation for  .
(c).[8%] Solve the recurrence relation obtained in part b. to find   .

5. [10%] Consider nine nonnegative real numbers   with sum 90.
Show that there must be four of the numbers having sum at least 40.

6. [8%] Classical experiments by E. Rutherford and H. Geiger in 1910 have shown that the number of alpha particles emitted per second in a radioactive process is a random variable X  having a Poisson distribution. If X  has mean 0.5, what is the probability of observing 2 or more particles during any given period?

7. Let X  be normal with mean -2 and variance 0.25. Determine c such that
(a).[4%] .
(b).[4%] 

Normal distribution with mean 0 and variance 1,

0.01 0.1 0.2 0.3 0.4 0.5
-2.326 -1.282 -0.842 -0.524 -0.253 0.000
0.65 0.75 0.85 0.95 0.99 0.999
0.385 0.674 1.036 1.645 2.326 3.090


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